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Full text of "Analytical Mechanics"

FIELDS OF FORCE AND NEWTONIAN POTENTIAL    2!
There are two cases which have to be considered separately.
(a) POINT OUTSIDE THE SPHERE.  Consider the sphere to be ma of concentric shells of thickness dp. Then, since the point is outside eve one of these shells the potential due to any one of the shells is, accordi to the results of the last problem,
,T7             dm
where dm is the mass of the shell and R the distance of the point from t'
center.   Hence the potential due to all the shells in the sphere is
\
dm
m
= _7_,
where m is the mass of the sphere. Therefore the potential at a poi outside of a sphere is the same as that due to a particle of equal m* placed at the center.
(b) POINT WITHIN THE SPHERE.  In this case we divide the sph< into two parts by means of a concentric spherical surface which pas* through the point. Then the potential due to that part of the sph< which is within the spherical surface is obtained by the result of case ( Thus if mi denotes the mass of this part of the sphere and Vi its potenti then
l     ~~       P
1
In order to find the potential due to the rest of the sphere suppose to be divided into a great number of concentric spherical shells. Th since every one of the shells contains the point the potential due to a one of them is
dVz =  7  =  47T7rp dp, P
where dm is the mass, p the radius, and dp the thickness of the sh< Therefore the potential due to all the shells having radii between R and c
P>a
=  47TYT (     O dp J R